Features of edge states and domain walls in chiral superconductors - - PowerPoint PPT Presentation

Features of edge states and domain walls in chiral superconductors

Manfred Sigrist kk

E

NQS2017, YITP Kyoto

Chiral superconductors

Chiral superconductors - candidates

Sr2RuO4 URu2Si2

tetragonal crystal structure µSR polar Kerr effect polar Kerr effect

• dd-parity

chiral p-wave chiral d-wave

even-parity

∆~

k = ∆0(kx ± iky)

∆~

k = ∆0(kx ± iky)kz

Lz = ±1

Chiral superconductors

Chiral superconductors - candidates

UPt3 SrPtAs

hexagonal crystal structure U Pt

• dd-parity

µSR polar Kerr effect µSR

even-parity

chiral f-wave chiral d-wave

∆~

k = ∆0(kx ± iky)2kz

∆~

k = ∆0(kx ± iky)2

Lz = ±2

Content

edge states and edge currents in a chiral p-wave SC

many other collaborators:

• Y. Imai, K. Wakabayashi,
• A. Furusaki, M. Matsumoto,
• C. Honerkamp, M. Fischer,

T.M. Rice, J. Goryo, W. Huang, ...

chiral domains

Sarah Etter Adrien Bouhon

former doctor students at ETH Zurich

focus on Sr2RuO4 as a chiral p-wave SC

Maeno et al 1994

Sr2RuO4

layered crystal structure quasi-2D metal

possible odd-parity spin-triplet states

ˆ Ψ(~ k) = ✓ kx ± iky kx ± iky ◆

ˆ Ψ(~ k) = ✓ −kx + iky kx + iky ◆

A-phase

chiral phase

B-phase

helical phase

ˆ Ψ = ✓ Ψ↑↑ Ψ↑↓ Ψ↓↑ Ψ↓↓ ◆

pair wave function

Maeno et al 1994

Sr2RuO4 - chiral p-wave superconductor

layered crystal structure quasi-2D metal

ˆ Ψ(~ k) = ✓ kx ± iky kx ± iky ◆

ˆ Ψ(~ k) = ✓ −kx + iky kx + iky ◆

A-phase

chiral phase

B-phase

helical phase identifaction

• intrinsic magnetism
• inplane spin polarizable
• multi-component
• polar Kerr effect
• phase-sensitive SQUID

Maeno et al 1994

Sr2RuO4 - chiral p-wave superconductor

layered crystal structure quasi-2D metal

A-phase

chiral phase identifaction

• intrinsic magnetism
• inplane spin polarizable
• multi-component
• polar Kerr effect
• phase-sensitive SQUID

ˆ Ψ(~ k) = ✓ ke±iθk ke±iθk ◆

phase winding around the FS

∆k = |∆0|e+iθk ∆k = |∆0|e−iθk

nodeless gap 2-fold degenerate chiral domains

gap function

chiral - broken time reversal symmetry

intrinsic magnetism in Sr2RuO4 ?

random local magnetism

Luke et al (1998)

µSR - zero-field relaxtion ''edge currents'' around inhomogeneities & defects muon-spin depolarization intrinsic magnetism

edge state currents

scanning probes at mesoscopic discs

T > Tc T < Tc

N SC SC-N disc

Mackenzie group (2014) R ≈ 5µm

scanning Hall probe

Edge currents

edge states for the chiral p-wave state

Sr2RuO4 - edge state spectrum

scattering of quasiparticles at the surface solution of Bogolyubov-de Gennes equations subgap bound states (close orbits in particle-hole space) “Andreev reflection”

surface

electron

hole

θ specular scattering

kk

phase shifts at turning points

k1 k2

for

φk1 + φk2 = π + θk2 − θk1

1 ~ I p · ds + φk1 + φk2 = 2πn

|p| ≈ E vF

θk2 − θk1 = π

E = 0

∆k = |∆0|eiθk

|E| ⌧ |∆0|

Bohr-Sommerfeld quantization

edge states for the chiral p-wave state

Sr2RuO4 - edge state spectrum

scattering of quasiparticles at the surface solution of Bogolyubov-de Gennes equations subgap bound states (close orbits in particle-hole space)

Bohr-Sommerfeld quantization

2θ = π − (θ~

k2 − θ~ k1) phase shifts “Andreev reflection”

E = Ekk = ∆0 sin θ = ∆0 kk kF

surface

electron

hole

θ specular scattering

kk

k1 k2

∆k = |∆0|eiθk

edge states for the chiral p-wave state

Sr2RuO4 - bulk and edge spectrum

scattering of quasiparticles at the surface solution of Bogolyubov-de Gennes equations subgap bound states (close orbits in particle-hole space)

Bohr-Sommerfeld quantization

2θ = π − (θ~

k2 − θ~ k1) phase shifts “Andreev reflection”

E = Ekk = ∆0 sin θ = ∆0 kk kF

kk

+kF −kF

E

+∆0

−∆0

continuum continuum

note: θ~

k2 − θ~ k1 = ±π

E = 0

result of topological property

Sr2RuO4 - bulk and edge spectrum

kk

+kF −kF

E

+∆0

−∆0

continuum continuum

• ccupied

surface

electron

hole c h a r g e spontaneous charge current

• rder parameter deformation

x

driving currents

Bz λ

screening currents

fields Bz ∼ 10 G

∆~

k = ηxkx + ηyky

|ηy|

|ηx|

|∆0|

Are edge currents a unique topological property?

lattice version of chiral p-wave superconductor (tight-binding):

⇠~

k = −2t(cos kxa + cos kya) + .... − ✏F

zeros of ∆~

k

+1-winding

• 1-winding

Chern numbers

kx

ky

1st Brillouin zone

N = +1

∆~

k = ∆0(sin kxa + i sin kya)

Topology and edge currents

Topology and edge currents

Are edge currents a unique topological property?

lattice version of chiral p-wave superconductor (tight-binding):

⇠~

k = −2t(cos kxa + cos kya) + .... − ✏F

zeros of ∆~

k

+1-winding

• 1-winding

Chern numbers

∆~

k = ∆0(sin kxa + i sin kya)

kx

ky

1st Brillouin zone

N = +1 − 4 × 1 2 = −1 × 1 = −1

kx

ky

kx

ky

kk

E

Topology and edge currents

kk

E

N = +1

N = −1

kk

E

kk

E

kx

ky

kx

ky

Topology and edge currents

N = +1

N = −1 vq = 1 ~ dEkk dkk

quasiparticle velocity

vq > 0 vq < 0

kx

ky

kx

ky

kk

E

Topology and edge currents

kk

E

N = +1

N = −1

surface

electron

hole c h a r g e

charge current

kk

• pposite chirality, but identical current direction

backward backward

Topology - thermal Hall effect

“Spontaneous” Righi-Leduc effect

chiral QP edge states

T1

heat current

chiral QP edge states

T2

temperature gradient

κxy = πk2

BT

12~ N + O(e−

∆ kBT )

Read & Green; Qin, Niu & Shi; Sumiyoshi & Fujimoto; …

Chern number 12~ κxy πk2

BT

µ

N = +1

N = −1

Lishitz-transition ( topological transition)

property of the (subgap) quasiparticles

T ⌧ Tc

Topology - thermal Hall effect

“Spontaneous” Righi-Leduc effect

chiral QP edge states

T1

chiral QP edge states

T2

temperature gradient

κxy = πk2

BT

12~ N + O(e−

∆ kBT )

Read & Green; Qin, Niu & Shi; Sumiyoshi & Fujimoto; …

Chern number 12~ κxy πk2

BT

µ

N = +1

N = −1

analog to ν=1 Quantum Hall effect

Lishitz-transition ( topological transition)

heat current

property of the (subgap) quasiparticles

T ⌧ Tc

vQP > 0

vQP < 0

Topology and edge currents - a further twist

∆~

k = ∆0(sin kxa + i sin kya)

particle-hole symmetry

∆~

k+ ~ Q = −∆~ k

~ Q = ⇡ a (1, 1)

kx ky

kk

E

x

y

~ n = (1, 0)

~ k1

~ k2

kk

E

Topology and edge currents - a further twist

∆~

k = ∆0(sin kxa + i sin kya)

particle-hole symmetry

∆~

k+ ~ Q = −∆~ k

~ Q = ⇡ a (1, 1)

kx ky

kk

E

x

y

~ n = (1, 0)

~ k1

~ k2

x

y

~ n = (1, 1)

kk

E

Topology and edge currents - a further twist

∆~

k = ∆0(sin kxa + i sin kya)

particle-hole symmetry

∆~

k+ ~ Q = −∆~ k

~ Q = ⇡ a (1, 1)

kx ky

kk

E

x

y

~ n = (1, 0)

~ k1

~ k2

x

y

~ n = (1, 1)

kk

E

Topology and edge currents - a further twist

∆~

k = ∆0(sin kxa + i sin kya)

particle-hole symmetry

∆~

k+ ~ Q = −∆~ k

~ Q = ⇡ a (1, 1)

kx ky

kk

E

x

y

~ n = (1, 0)

~ k1

~ k2

x

y

~ n = (1, 1)

~ Q

θ~

k2 − θ~ k1 = ±π

E = 0

new zero-energy momenta

no change in Chern number

Topology and edge currents - a further twist

kk

E

~ n = (1, 1)

kk

E

kx ky

~ k1

~ k2

~ Q

kx ky

~ k1

~ k2

current reversal

Topology and edge currents - a further twist

kk

E

~ n = (1, 1)

kk

E

current reversal

x

y

x

y

circular supercurrent non-circular supercurrent

• A. Bouhon

Ginzburg-Landau approach - chiral p-wave

F = Z dV

• a|η|2 + b1|η|4 + b2(η∗2

x η2 y + η2 xη∗2 y ) + b3|ηx|2|ηy|2

+K1(|Πxηx|2 + |Πyηy|2) + K2(|Πxηy|2 + |Πyηx|2) + [K3(Πxηx)∗(Πyηy) + K4(Πxηy)∗(Πyηx) + c.c.] +K5|Πzη|2 + (r × A)/8π

• rder parameter:

d(k) = ˆ zη · k = ˆ z(ηxkx + ηyky)

Π = ~ i r + 2e c A

and

Ginzburg-Landau approach - chiral p-wave

F = Z dV

• a|η|2 + b1|η|4 + b2(η∗2

x η2 y + η2 xη∗2 y ) + b3|ηx|2|ηy|2

+K1(|Πxηx|2 + |Πyηy|2) + K2(|Πxηy|2 + |Πyηx|2) + [K3(Πxηx)∗(Πyηy) + K4(Πxηy)∗(Πyηx) + c.c.] +K5|Πzη|2 + (r × A)/8π

• rder parameter:

d(k) = ˆ zη · k = ˆ z(ηxkx + ηyky)

length scales for amplitude modulations for the two order parameter components

Π = ~ i r + 2e c A

and

Ginzburg-Landau approach - chiral p-wave

F = Z dV

• a|η|2 + b1|η|4 + b2(η∗2

x η2 y + η2 xη∗2 y ) + b3|ηx|2|ηy|2

+K1(|Πxηx|2 + |Πyηy|2) + K2(|Πxηy|2 + |Πyηx|2) + [K3(Πxηx)∗(Πyηy) + K4(Πxηy)∗(Πyηx) + c.c.] +K5|Πzη|2 + (r × A)/8π

• rder parameter:

d(k) = ˆ zη · k = ˆ z(ηxkx + ηyky)

edge currents for chiral phase

Π = ~ i r + 2e c A

and

Ginzburg-Landau approach - chiral p-wave

F = Z dV

• a|η|2 + b1|η|4 + b2(η∗2

x η2 y + η2 xη∗2 y ) + b3|ηx|2|ηy|2

+K1(|Πxηx|2 + |Πyηy|2) + K2(|Πxηy|2 + |Πyηx|2) + [K3(Πxηx)∗(Πyηy) + K4(Πxηy)∗(Πyηx) + c.c.] +K5|Πzη|2 + (r × A)/8π

• rder parameter:

d(k) = ˆ zη · k = ˆ z(ηxkx + ηyky)

cylindrically symmetric bands

1 3K1 = K2 = K3 = K4 K5

Π = ~ i r + 2e c A

and

Ginzburg-Landau approach - edge currents

x y

K1|Πxηx|2 + K2|Πxηy|2

length scales of order parameter components

ξ2

x/ξ2 y = K1/K2

current density along edge:

jy = 16πe~ ✓ K3|ηx| ∂ ∂x|ηy| − K4|ηy| ∂ ∂x|ηx| ◆ + cλ−2 4π Ay driving current screening current

note: length scales important

sample edge

Ginzburg-Landau approach - edge currents

K1(|Πxηx|2 + |Πyηy|2) + K2(|Πxηy|2 + |Πyηx|2)

kx ky

~ k1

~ k2

~ Q

kx ky

~ k1

~ k2

K1 > K2 K1 < K2

x

y x y

determines current patter

(specular scattering)

Ginzburg-Landau approach - edge currents

K1(|Πxηx|2 + |Πyηy|2) + K2(|Πxηy|2 + |Πyηx|2)

kx ky

~ k1

~ k2

~ Q

K1 < K2

x

y

γ

• band is most likely dominant

and crosses Umklapp diamond

γ

Disk shaped sample

Ginzburg-Landau approach - edge currents

disk shaped sample

• S. Etter

current and flux pattern for all surface orientations

~ n

• specular scattering

η⊥

ηk

component perpendicular to surface R = 40ξ0

isotropic

component parallel to surface

Bz(r, θ)

θ r

• r/ξ0

η η

• r/ξ0

Bz(γξ0

2)

• ξ

θ

πγξ

Ginzburg-Landau approach - edge currents

different boundary conditions

• specular scattering

totally pair breaking diffuse scattering

η⊥

ηk

η⊥

ηk

η⊥

ηk

rotation symmetric

• ξ

η η

• r/ξ0

γξ

• r/ξ0

Jθ (4πγξ0

3/c)

K1 K2 = 3

Bz(r, θ)

Ginzburg-Landau approach - edge currents

boundary conditions specular pair breaking diffuse

Anisotropy

isotropic

scanning anisotropy and boundary conditions

Bz(r, θ)

Ginzburg-Landau approach - edge currents

boundary conditions specular pair breaking diffuse

Anisotropy

isotropic

scanning anisotropy and boundary conditions

Bz(r, θ)

Ginzburg-Landau approach - edge currents

boundary conditions specular pair breaking diffuse

Anisotropy

isotropic

scanning anisotropy and boundary conditions

maximal fluxes

Furusaki, Matsumoto & MS

Bz(r, θ)

Ginzburg-Landau approach - edge currents

boundary conditions specular

Anisotropy

isotropic

scanning anisotropy and boundary conditions

Bz(r, θ)

Ginzburg-Landau approach - edge currents

boundary conditions specular

Anisotropy

isotropic

scanning anisotropy and boundary conditions

net flux in disk vanishes

Chiral domain domain walls

Chiral domains and domain walls - chiral p-wave chiral p-wave state

discrete degeneracy 2 2 types of domains in-plane domain walls

x y

x

ηx

−iηy

x

ηx

−iηy

Type 1 Type 2 phase switch ~ 0 phase switch ~ π

+ -

∆+

k = ˆ

z∆0(kx + iky) ∆−

k = ˆ

z∆0(kx − iky)

∆k = ηxkx + ηyky

+ -

Chiral domains and domain walls - chiral p-wave

“parabolic band rotation symmetric” γ-band like Fermi surface in BZ

kx ky

~ k1

~ k2

~ Q

kx ky

~ k1

~ k2

γ

K1 > K2

K1 < K2

x y

+ -

s-wave SC “parabolic band rotation symmetric” γ-band like Fermi surface in BZ

π - loop

Chiral domains and domain walls - chiral p-wave

x y

Jy ∝ Im(η∗

s(n × η)z) = |ηs||ηx| sin(φs − φx)

Josephson coupling

conservation of total angular momentum coupling with the x-component

φx

π

changes by through the domain wall intrinsic phase twist

+ -

s-wave SC “parabolic band rotation symmetric” γ-band like Fermi surface in BZ

narrow s-p-junctions

Sr2RuO4-Cu-PbIn

Bahr & van Harlingen PbIn 500 nm Sr2RuO4

π - loop interference pattern

Chiral domains and domain walls - chiral p-wave

+ -

+ -

s-wave SC “parabolic band rotation symmetric” γ-band like Fermi surface in BZ

π-loop

Chiral domains and domain walls - chiral p-wave

+ -

s-wave SC

extended interface many domains

van Harlingen group

+ -

+ -

s-wave SC “parabolic band rotation symmetric” γ-band like Fermi surface in BZ

π-loop

Chiral domains and domain walls - chiral p-wave

+ -

s-wave SC

extended interface many domains

van Harlingen group

irregular – history dependent interference I(H) = maxφ Z dx jc sin ✓ φ − φx(x) − 2πHd Φ0 x ◆

Interference pattern in a magnetic field

Conclusions

edge currents “invisible”

• supercurrents are not universal
• vanishing would be however accidential
• note: quantum thermal Hall effect universal

domain walls

• defects of condensate history dependence
• visible through interference effects

see also C. Kallin group